library(readr)
library(ggplot2)
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(tidyverse)
## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2
## ──
## ✔ tibble  3.1.8     ✔ stringr 1.5.0
## ✔ tidyr   1.3.0     ✔ forcats 1.0.0
## ✔ purrr   1.0.1     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
library(grid)
library(cowplot) 
library(factoextra)
## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa
library(corrplot)
## corrplot 0.92 loaded
library(devtools)
## Loading required package: usethis
library(FactoMineR)
library(ggfortify)
library(psych)
## 
## Attaching package: 'psych'
## 
## The following objects are masked from 'package:ggplot2':
## 
##     %+%, alpha
library(vctrs)
## 
## Attaching package: 'vctrs'
## 
## The following object is masked from 'package:tibble':
## 
##     data_frame
## 
## The following object is masked from 'package:dplyr':
## 
##     data_frame
library(purrr)
library("corrplot")

tita <- read_csv("/Users/asrithanarra/Documents/Multivariate analysis/Titanic.csv")
## Rows: 891 Columns: 5
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (1): Survived
## dbl (4): PassengerId, Pclass, Age, Fare
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
str(tita)
## spc_tbl_ [891 × 5] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
##  $ Survived   : chr [1:891] "No" "yes" "yes" "yes" ...
##  $ PassengerId: num [1:891] 1 2 3 4 5 6 7 8 9 10 ...
##  $ Pclass     : num [1:891] 3 1 3 1 3 3 1 3 3 2 ...
##  $ Age        : num [1:891] 22 38 26 35 35 10 54 2 27 14 ...
##  $ Fare       : num [1:891] 7.25 71.28 7.92 53.1 8.05 ...
##  - attr(*, "spec")=
##   .. cols(
##   ..   Survived = col_character(),
##   ..   PassengerId = col_double(),
##   ..   Pclass = col_double(),
##   ..   Age = col_double(),
##   ..   Fare = col_double()
##   .. )
##  - attr(*, "problems")=<externalptr>
attach(tita)
summary(tita)
##    Survived          PassengerId        Pclass           Age       
##  Length:891         Min.   :  1.0   Min.   :1.000   Min.   : 1.00  
##  Class :character   1st Qu.:223.5   1st Qu.:2.000   1st Qu.:22.00  
##  Mode  :character   Median :446.0   Median :3.000   Median :28.00  
##                     Mean   :446.0   Mean   :2.309   Mean   :30.32  
##                     3rd Qu.:668.5   3rd Qu.:3.000   3rd Qu.:38.00  
##                     Max.   :891.0   Max.   :3.000   Max.   :92.00  
##       Fare        
##  Min.   :  0.000  
##  1st Qu.:  7.925  
##  Median : 15.100  
##  Mean   : 33.389  
##  3rd Qu.: 31.275  
##  Max.   :512.329
cor(tita[-1])
##               PassengerId      Pclass         Age          Fare
## PassengerId  1.0000000000 -0.03514399  0.04815802  0.0008510998
## Pclass      -0.0351439940  1.00000000 -0.30758636 -0.5431017648
## Age          0.0481580247 -0.30758636  1.00000000  0.0842967291
## Fare         0.0008510998 -0.54310176  0.08429673  1.0000000000
plot(tita[,-1])

library(GGally)
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
ggpairs(tita[,c( "Survived","PassengerId", "Pclass", "Age","Fare")],aes(color = factor(Survived)))
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

p1 <- ggplot(tita, aes(x=PassengerId, y=Pclass, color=Survived)) + geom_point()
p2 <- ggplot(tita, aes(x=Age, y=Fare, color=Survived)) + geom_point()
p3 <- ggplot(tita, aes(x=Age, y=PassengerId, color=Survived)) + geom_point()
p4 <- ggplot(tita, aes(x=Pclass, y=Fare, color=Survived)) + geom_point()
plot_grid(p1, p2, p3, p4, labels = "AUTO")

tita_pca <- prcomp(tita[,-1],scale=TRUE)
tita_pca
## Standard deviations (1, .., p=4):
## [1] 1.2905711 1.0074002 0.9542240 0.6395527
## 
## Rotation (n x k) = (4 x 4):
##                     PC1         PC2         PC3        PC4
## PassengerId  0.06565727 -0.90999815 -0.40894567 0.01886594
## Pclass      -0.68367669 -0.05910479  0.05521554 0.72528894
## Age          0.39775028 -0.32819164  0.80738894 0.28671950
## Fare         0.60833381  0.24637418 -0.42170818 0.62561326
summary(tita_pca)
## Importance of components:
##                           PC1    PC2    PC3    PC4
## Standard deviation     1.2906 1.0074 0.9542 0.6396
## Proportion of Variance 0.4164 0.2537 0.2276 0.1023
## Cumulative Proportion  0.4164 0.6701 0.8977 1.0000
plot(tita_pca)

tita_pca
## Standard deviations (1, .., p=4):
## [1] 1.2905711 1.0074002 0.9542240 0.6395527
## 
## Rotation (n x k) = (4 x 4):
##                     PC1         PC2         PC3        PC4
## PassengerId  0.06565727 -0.90999815 -0.40894567 0.01886594
## Pclass      -0.68367669 -0.05910479  0.05521554 0.72528894
## Age          0.39775028 -0.32819164  0.80738894 0.28671950
## Fare         0.60833381  0.24637418 -0.42170818 0.62561326
(eigen_tita <- tita_pca$sdev^2)
## [1] 1.6655736 1.0148552 0.9105435 0.4090277
names(eigen_tita) <- paste("PC",1:4,sep="")
eigen_tita
##       PC1       PC2       PC3       PC4 
## 1.6655736 1.0148552 0.9105435 0.4090277
sumlabdas<-sum(eigen_tita)
sumlabdas
## [1] 4
propvar<-eigen_tita/sumlabdas
propvar
##       PC1       PC2       PC3       PC4 
## 0.4163934 0.2537138 0.2276359 0.1022569
cumvar<-cumsum(propvar)
cumvar
##       PC1       PC2       PC3       PC4 
## 0.4163934 0.6701072 0.8977431 1.0000000
matlambdas <- rbind(eigen_tita,propvar,cumvar)
rownames(matlambdas) <- c("Eigenvalues","Prop. variance","Cum. prop. variance")
round(matlambdas,4)
##                        PC1    PC2    PC3    PC4
## Eigenvalues         1.6656 1.0149 0.9105 0.4090
## Prop. variance      0.4164 0.2537 0.2276 0.1023
## Cum. prop. variance 0.4164 0.6701 0.8977 1.0000
summary(tita_pca)
## Importance of components:
##                           PC1    PC2    PC3    PC4
## Standard deviation     1.2906 1.0074 0.9542 0.6396
## Proportion of Variance 0.4164 0.2537 0.2276 0.1023
## Cumulative Proportion  0.4164 0.6701 0.8977 1.0000
tita_pca$rotation
##                     PC1         PC2         PC3        PC4
## PassengerId  0.06565727 -0.90999815 -0.40894567 0.01886594
## Pclass      -0.68367669 -0.05910479  0.05521554 0.72528894
## Age          0.39775028 -0.32819164  0.80738894 0.28671950
## Fare         0.60833381  0.24637418 -0.42170818 0.62561326
tita_pca$x
##                 PC1           PC2          PC3           PC4
##   [1,] -1.224789952  1.591832e+00  0.493032238  0.0777818918
##   [2,]  1.625007860  1.666786e+00  0.745451365 -0.5472209837
##   [3,] -1.103520752  1.495017e+00  0.713072412  0.1674523980
##   [4,]  1.324249082  1.641688e+00  0.720920576 -0.8309019381
##   [5,] -0.847915525  1.279294e+00  1.223651608  0.3519432306
##   [6,] -1.547252191  1.858991e+00 -0.211283668 -0.1507895692
##   [7,]  1.845652645  1.183347e+00  1.813157746 -0.4599148610
##   [8,] -1.621785174  2.098829e+00 -0.776301561 -0.1584874252
##   [9,] -1.035570658  1.466038e+00  0.734227701  0.2275319110
##  [10,] -0.358184922  1.926870e+00 -0.233464501 -0.6718178975
##  [11,] -1.616810493  2.020600e+00 -0.630521362 -0.1712712149
##  [12,]  1.657931836  9.504729e-01  2.243158923 -0.6885782840
##  [13,] -1.268549617  1.599763e+00  0.352955750  0.0478430692
##  [14,] -0.456078004  1.266583e+00  1.246243564  0.7185439870
##  [15,] -1.439443222  1.731249e+00  0.008202102 -0.0762850737
##  [16,]  0.630942254  8.844586e-01  2.218419701 -0.0110473235
##  [17,] -1.523537891  2.105865e+00 -0.857118035 -0.0591510449
##  [18,] -0.334190889  1.630171e+00  0.352466086 -0.7179851672
##  [19,] -0.838459118  1.370824e+00  0.890395104  0.3936865149
##  [20,] -1.220240572  1.524528e+00  0.463047037  0.0788682829
##  [21,]  0.187845082  1.380062e+00  0.983868989 -0.2943499693
##  [22,]  0.004969680  1.337021e+00  1.032496751 -0.4739426413
##  [23,] -1.407137992  1.680556e+00  0.051242699 -0.0532410295
##  [24,]  0.922321774  1.648761e+00  0.434171789 -1.1873630545
##  [25,] -1.448378021  1.899214e+00 -0.460121836 -0.0353665516
##  [26,] -0.479853916  1.247945e+00  1.169046617  0.7004902552
##  [27,] -1.218454700  1.499776e+00  0.451923754  0.0793814346
##  [28,]  3.381400508  2.942091e+00 -1.966749325  1.4188051166
##  [29,] -1.210146780  1.495862e+00  0.443340184  0.0875472097
##  [30,] -1.181515454  1.469156e+00  0.498812884  0.1081364407
##  [31,]  1.169524327  1.307450e+00  1.173712890 -1.0384577635
##  [32,]  2.106772294  2.272660e+00 -0.781871774  0.0725502181
##  [33,] -1.182487929  1.457844e+00  0.495250471  0.1065691516
##  [34,]  0.879939532  5.385050e-01  2.864450003  0.1462902599
##  [35,]  1.481416034  1.835161e+00  0.031063354 -0.6144676657
##  [36,]  1.516550102  1.360473e+00  1.079552617 -0.6998527733
##  [37,] -1.215853393  1.464437e+00  0.435998645  0.0801659919
##  [38,] -1.233993165  1.488113e+00  0.370428640  0.0699881951
##  [39,] -1.199675038  1.602361e+00  0.115028565  0.1310909247
##  [40,] -1.392688145  1.659202e+00 -0.059513964 -0.0329285345
##  [41,] -0.680854202  1.042625e+00  1.440666248  0.4736121025
##  [42,] -0.091821012  1.467674e+00  0.534221654 -0.5165998675
##  [43,] -1.178198836  1.423188e+00  0.478155358  0.1090894366
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##  [45,] -1.290599806  1.509038e+00  0.246318830  0.0277828037
##  [46,] -1.175595491  1.413325e+00  0.472114121  0.1111995369
##  [47,] -0.522974058  9.807421e-01  1.552945664  0.6088435068
##  [48,] -0.868699330  1.149049e+00  1.100602789  0.3311056173
##  [49,] -0.730594851  1.236004e+00  0.926721397  0.4816101218
##  [50,] -1.199252548  1.562500e+00  0.099201669  0.1294457130
##  [51,] -1.248073308  1.920378e+00 -0.712426160  0.1743783624
##  [52,] -1.233401272  1.437403e+00  0.350247760  0.0679500073
##  [53,]  2.012892797  1.256984e+00  1.248600133 -0.2532898036
##  [54,]  0.027194156  1.402878e+00  0.588237240 -0.4138054740
##  [55,]  2.288445338  8.067009e-01  2.282479999 -0.1089491027
##  [56,]  1.099555794  1.396107e+00  0.726515907 -1.0631425701
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##  [75,] -0.337147003  1.335391e+00  0.540526702  0.8899843177
##  [76,] -1.116352837  1.258813e+00  0.542145810  0.1491204553
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##  [79,]  1.590960785  7.343304e-02  3.612463709  0.7216729377
##  [80,] -0.916929542  1.151709e+00  0.781916407  0.3101205701
##  [81,] -1.183521043  1.317402e+00  0.351449193  0.1050979201
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##  [95,] -0.158209696  3.991814e-01  2.460021815  0.8362331162
##  [96,] -0.907148317  1.028116e+00  0.906006237  0.2998220220
##  [97,]  2.142533023  3.867797e-01  2.784713261 -0.3189444233
##  [98,]  1.132363486  1.637832e+00 -0.199597789 -0.9420144639
##  [99,]  0.143808258  1.113024e+00  0.827513169 -0.3457183283
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## [101,] -1.022509918  1.101849e+00  0.671985521  0.2149034715
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## [802,]  0.277363817 -1.287330e+00 -0.488032922 -0.3152821790
## [803,]  1.649221221 -3.025992e-01 -2.474274164 -0.4397701112
## [804,] -0.441259883 -1.706455e+00  0.350544356  0.5584237097
## [805,] -0.882055985 -1.368676e+00 -0.496289542  0.2349122928
## [806,] -0.759551984 -1.461352e+00 -0.275693171  0.3260417372
## [807,]  1.271134109 -1.440836e+00 -0.069311904 -1.0720090402
## [808,] -1.125360145 -1.166168e+00 -1.022456981  0.0621266142
## [809,]  0.346644365 -1.562048e+00  0.067916404 -0.3146875255
## [810,]  1.473522758 -1.161811e+00 -0.674243856 -0.8124410717
## [811,] -0.897827125 -1.362236e+00 -0.570562443  0.2262250883
## [812,] -0.337414363 -1.589524e+00  0.037061330  0.6897052804
## [813,]  0.205153009 -1.495258e+00 -0.146578500 -0.4262889726
## [814,] -1.181863591 -7.949349e-01 -1.912552594  0.1068793269
## [815,] -0.782156368 -1.468598e+00 -0.349465693  0.3097600025
## [816,]  1.708951874 -7.763343e-01 -1.536514551 -0.4934624534
## [817,] -0.980384419 -1.313520e+00 -0.752003246  0.1661872822
## [818,]  0.409629411 -1.291991e+00 -0.602316798 -0.1822846596
## [819,] -0.433888498 -1.792722e+00  0.400984269  0.5545022267
## [820,] -1.107847459 -9.254451e-01 -1.665404458  0.1471983066
## [821,]  2.493261296 -1.447442e+00  0.061240882  0.0695234595
## [822,] -0.857604882 -1.420642e+00 -0.537246615  0.2568438333
## [823,]  2.057556899 -1.145903e+00 -0.713802217 -0.2576069804
## [824,] -0.811651924 -1.409309e+00 -0.571926417  0.3037239376
## [825,] -1.191498634 -7.002187e-01 -2.228338003  0.1295560686
## [826,] -1.017888051 -1.326800e+00 -0.815447335  0.1345830919
## [827,] -0.427076764 -1.091161e+00 -1.226420761  0.7419870546
## [828,] -0.433169522 -6.298359e-01 -2.334174270 -0.7909248267
## [829,] -1.007587159 -1.333546e+00 -0.826824653  0.1446093856
## [830,]  2.616428922 -1.776940e+00  0.730475618  0.1078251194
## [831,] -1.124415249 -1.145501e+00 -1.285790155  0.0847487561
## [832,]  1.660895701 -2.638640e+00  2.500609379  0.6512292600
## [833,] -0.082889088 -2.117471e+00  1.058686197  0.8088292323
## [834,] -0.976891195 -1.373973e+00 -0.778431932  0.1665656438
## [835,] -1.112214099 -1.259105e+00 -1.069699016  0.0705413449
## [836,]  2.007502904 -1.248148e+00 -0.620729609 -0.3202068585
## [837,] -0.064784515 -2.124695e+00  1.040487041  0.8266918023
## [838,] -0.691753486 -1.619677e+00 -0.214416917  0.3723833964
## [839,]  0.505869925 -1.900860e+00  0.642074527  1.4131773083
## [840,]  1.202262555 -1.380850e+00 -0.528566802 -1.0970782185
## [841,] -1.058796448 -1.328632e+00 -0.961736924  0.1070093352
## [842,] -0.322836827 -1.156042e+00 -1.279439817 -0.8100994445
## [843,]  1.133988130 -1.315431e+00 -0.715672201 -1.1418602667
## [844,] -0.681264431 -1.671927e+00 -0.153428579  0.3733697144
## [845,] -1.133520411 -1.269464e+00 -1.145783568  0.0554054894
## [846,] -0.442067137 -1.859633e+00  0.291792253  0.5496528455
## [847,]  0.015407338 -1.331369e+00 -0.794076094  1.1065955355
## [848,] -0.634683534 -1.702282e+00 -0.114635402  0.4118511748
## [849,]  0.285275595 -1.351185e+00 -0.790088070 -0.2900327961
## [850,]  1.574736096 -8.504398e-01 -1.721685880 -0.6119198689
## [851,] -1.228780663 -8.792652e-01 -2.085744896  0.0689668172
## [852,]  0.463852331 -2.623779e+00  2.110763763  1.2028488555
## [853,] -1.278437149 -1.079968e+00 -1.670483249 -0.0258097250
## [854,]  0.842420702 -9.882701e-01 -1.603343348 -1.3224611632
## [855,]  0.654223977 -1.778200e+00  0.173399349 -0.0503996412
## [856,] -1.094341115 -1.328292e+00 -1.111744750  0.0849516626
## [857,]  3.155845582 -1.067473e+00 -0.986041843  0.8047859077
## [858,]  1.676518796 -1.878213e+00  0.498438361 -0.7687473291
## [859,] -0.806404724 -1.430572e+00 -0.855188229  0.3285018221
## [860,] -1.315847654 -1.189920e+00 -1.500969594 -0.0829388885
## [861,] -0.388247613 -1.857763e+00  0.156568173  0.6108314231
## [862,] -0.164923241 -1.338187e+00 -1.033488862 -0.6948131255
## [863,]  1.585859838 -1.829138e+00  0.324025965 -0.8369278606
## [864,] -0.543822331 -9.264707e-01 -1.965067827  0.7015929337
## [865,] -0.061743751 -1.411305e+00 -0.879053396 -0.6152689326
## [866,]  0.445721480 -1.833350e+00  0.148937804 -0.2495716807
## [867,]  0.033531936 -1.483985e+00 -0.717726686 -0.5436639829
## [868,]  1.400922802 -1.332968e+00 -0.859288164 -0.8807363173
## [869,] -1.230128283 -1.257283e+00 -1.419636199 -0.0138188536
## [870,] -1.464010257 -1.043680e+00 -1.949510904 -0.1765365852
## [871,] -0.882420724 -1.574355e+00 -0.665973453  0.2307252723
## [872,]  1.877331620 -1.709184e+00  0.032530073 -0.5302122319
## [873,]  0.916272507 -1.616773e+00 -0.376915299 -1.3974307970
## [874,] -0.276748825 -2.067893e+00  0.521312648  0.6710417075
## [875,]  0.184634241 -1.486566e+00 -0.757038404 -0.3984484843
## [876,] -1.199102366 -1.339518e+00 -1.297563930 -0.0005676884
## [877,] -1.026717184 -1.446655e+00 -1.034813465  0.1331934991
## [878,] -1.077883227 -1.436354e+00 -1.077489052  0.0890513344
## [879,] -1.246698138 -1.300386e+00 -1.422271506 -0.0327500064
## [880,]  2.497760150 -1.798990e+00  0.281733948  0.0283301695
## [881,]  0.125468768 -1.428376e+00 -0.954694846 -0.4344300352
## [882,] -0.682365979 -1.776005e+00 -0.283060583  0.3737187430
## [883,] -0.960833034 -1.511133e+00 -0.935493380  0.1824820944
## [884,]  0.026018474 -1.583559e+00 -0.659792690 -0.5632712381
## [885,] -0.917108748 -1.604691e+00 -0.738430291  0.2010713479
## [886,] -0.259236139 -1.827171e+00 -0.121634829  0.7561133998
## [887,]  0.028404006 -1.558848e+00 -0.742415580 -0.5527188461
## [888,]  0.823587602 -1.223622e+00 -1.408104200 -1.3742562540
## [889,] -1.058750176 -1.260661e+00 -1.566682323  0.1586458981
## [890,]  1.021346226 -1.393447e+00 -1.010889966 -1.2319225497
## [891,] -0.709986046 -1.785282e+00 -0.353356141  0.3522788569
var <- get_pca_var(tita_pca)
var
## Principal Component Analysis Results for variables
##  ===================================================
##   Name       Description                                    
## 1 "$coord"   "Coordinates for the variables"                
## 2 "$cor"     "Correlations between variables and dimensions"
## 3 "$cos2"    "Cos2 for the variables"                       
## 4 "$contrib" "contributions of the variables"
plot(eigen_tita, xlab = "Component number", ylab = "Component variance", type = "l", main = "Scree diagram")

pairs.panels(tita[-1],
             gap = 0,
             bg = c("red", "blue")[tita$Survived],
             pch=21)

diag(cov(tita_pca$x))
##       PC1       PC2       PC3       PC4 
## 1.6655736 1.0148552 0.9105435 0.4090277
xlim <- range(tita_pca$x[,1])
plot(tita_pca$x,xlim=xlim,ylim=xlim)

#Correlation circle
head(var$coord, 4)
##                   Dim.1       Dim.2      Dim.3      Dim.4
## PassengerId  0.08473538 -0.91673233 -0.3902258 0.01206576
## Pclass      -0.88233335 -0.05954218  0.0526880 0.46386051
## Age          0.51332500 -0.33062032  0.7704299 0.18337223
## Fare         0.78509800  0.24819741 -0.4024041 0.40011266
fviz_eig(tita_pca, addlabels = TRUE)

fviz_pca_var(tita_pca,col.var = "cos2",
             gradient.cols = c("#FFCC00", "#CC9933", "#660033", "#330033"),
             repel = TRUE)

biplot(tita_pca)

head(var$cos2, 4)
##                   Dim.1       Dim.2       Dim.3        Dim.4
## PassengerId 0.007180084 0.840398167 0.152276167 0.0001455826
## Pclass      0.778512135 0.003545271 0.002776025 0.2151665693
## Age         0.263502553 0.109309798 0.593562274 0.0336253747
## Fare        0.616378869 0.061601952 0.161929040 0.1600901383
corrplot(var$cos2, is.corr=FALSE)

fviz_pca_var(tita_pca, col.var = "cos2",
             gradient.cols = c("#00FF00", "#FF0000", "#FC4E07"), 
             repel = TRUE 
)

#Contributions of  Pc
head(var$contrib, 4)
##                  Dim.1      Dim.2      Dim.3       Dim.4
## PassengerId  0.4310877 82.8096636 16.7236563  0.03559235
## Pclass      46.7413818  0.3493376  0.3048756 52.60440497
## Age         15.8205285 10.7709750 65.1876896  8.22080696
## Fare        37.0070019  6.0700239 17.7837785 39.13919571
corrplot(var$contrib, is.corr=FALSE)   

fviz_pca_var(tita_pca, col.var = "contrib",
             gradient.cols = c("#00FF00", "#E7B800", "#FC4E07")
)

# Contributions of  PC1
fviz_contrib(tita_pca, choice = "var", axes = 1, top = 10)

# Contributions of  PC2
fviz_contrib(tita_pca, choice = "var", axes = 2, top = 10)

fviz_contrib(tita_pca, choice = "var", axes = 1:2, top = 10)

#Graphs of ind

ind <- get_pca_ind(tita_pca)
ind
## Principal Component Analysis Results for individuals
##  ===================================================
##   Name       Description                       
## 1 "$coord"   "Coordinates for the individuals" 
## 2 "$cos2"    "Cos2 for the individuals"        
## 3 "$contrib" "contributions of the individuals"
# Coordinates of individuals
head(ind$coord)
##        Dim.1    Dim.2      Dim.3       Dim.4
## 1 -1.2247900 1.591832  0.4930322  0.07778189
## 2  1.6250079 1.666786  0.7454514 -0.54722098
## 3 -1.1035208 1.495017  0.7130724  0.16745240
## 4  1.3242491 1.641688  0.7209206 -0.83090194
## 5 -0.8479155 1.279294  1.2236516  0.35194323
## 6 -1.5472522 1.858991 -0.2112837 -0.15078957
# Quality of individuals
head(ind$cos2)
##       Dim.1     Dim.2       Dim.3       Dim.4
## 1 0.3502336 0.5916014 0.056752524 0.001412510
## 2 0.4208897 0.4428094 0.088571904 0.047729055
## 3 0.3052526 0.5602611 0.127457556 0.007028798
## 4 0.3098898 0.4762657 0.091842312 0.122002158
## 5 0.1807915 0.4115412 0.376520245 0.031147127
## 6 0.4045803 0.5840329 0.007544221 0.003842600
# Contributions of individuals
head(ind$contrib)
##        Dim.1     Dim.2       Dim.3       Dim.4
## 1 0.10108383 0.2802289 0.029962099 0.001660071
## 2 0.17793828 0.3072399 0.068495213 0.082166559
## 3 0.08205772 0.2471783 0.062674209 0.007694003
## 4 0.11816744 0.2980572 0.064061402 0.189438897
## 5 0.04844664 0.1809917 0.184559823 0.033987158
## 6 0.16131720 0.3821844 0.005502416 0.006238960
fviz_pca_ind(tita_pca, col.ind = "cos2", 
             gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
             repel = TRUE # Avoid text overlapping (slow if many points)
)

fviz_pca_ind(tita_pca,
             geom.ind = "point",
             col.ind = tita$Survived, # color by groups
             palette = c("#00FF00", "#E7B800", "#FC4E07"),
             addEllipses = TRUE, # ellipses
             legend.title = "Survivorship"
)

ind.p <- fviz_pca_ind(tita_pca, geom = "point", col.ind = tita$Survived)
ggpubr::ggpar(ind.p,
              title = "Principal Component Analysis",
              subtitle = "titanic",
              xlab = "PC1", ylab = "PC2",
              legend.title = "Survivorship", legend.position = "top",
              ggtheme = theme_gray(), palette = "jco"
)

#anova testing
tita_anova <- aov(PassengerId+Pclass ~ Survived, data = tita)
summary(tita_anova)
##              Df   Sum Sq Mean Sq F value Pr(>F)
## Survived      1     2198    2198   0.033  0.856
## Residuals   889 58930554   66289